Functions

In grade 8, students are first introduced to the notion of functions. They understand a function as a rule that assigns to each input exactly one output 8.F.A.1. The main focus is on  • linear functions and their representations (equations, graphs, tables, or verbal descriptions);  • understanding that the equation y = mx + b defines a linear function, whose graph is a line;  • modeling a linear relationship by a function, determining rate of change and initial value and interpreting them in terms of a situation modeled by the function and in terms of its graph or a table of values;  • describing qualitatively a functional relationship between two quantities by analyzing a graph, e.g., where it is increasing or decreasing, linear or nonlinear. In this unit, students begin to use formal notation for functions, writing equations such as $f(x) = 2x + 3$ to describe a function. Students develop the understanding that the input/output relationship is a correspondence between two sets, and use the terms domain and range to describe them. They develop fluency with function notation and its use in describing qualitative features of the graph of a function by first interpreting, then writing expressions, equations, and inequalities such as $f(x + 2)$, $f(a) = 2$, $f(x) > 2$, and $f(x) > g(x).$ Students expand their repertoires of functions, working with piecewise-defined functions, including step functions. Building on their experiences with rate of change and slope from grade 8, students examine the behavior of non-linear functions. They describe key aspects of their graphs, and calculate and interpret average rates of change over specified intervals. Students’ work with domain and range provides a basis for understanding when a function has an inverse. They examine simple functions that do and do not have inverses and write expressions for inverses of linear functions, but do not draw general conclusions about when a function has an inverse. Functions are a unifying theme in high school mathematics. In statistics, functions play especially prominent supporting roles as lines of best fit for bivariate statistics (units S2 and S4) and the normal distribution (unit S4). In geometry, transformations are viewed as functions sending points in the plane to points in the plane (units G1, G2, and G3), and ratios of sides of right triangles lead to the trigonometric functions on acute angles (unit G4 and A9). In algebra, students study systems of linear equations and inequalities (unit A1), exponential functions and geometric sequences (units A2 and A6), quadratic functions (unit A3), and rational functions, polynomials, and logarithms (unit A7 and A8).